The problem asks for the coordinates of the image of point $X$ after two transformations: a reflection over the $y$-axis, followed by a $180^\circ$ clockwise rotation. Let's solve step by step:

Step 1: Identify point $X$

From the graph, point $X$ is located at $(4, 4)$.

Step 2: Reflect over the $y$-axis

To reflect over the $y$-axis, we negate the $x$-coordinate of $X$. The new coordinates become:

$X' = (-4, 4)$

Step 3: Rotate $180^\circ$ clockwise

A $180^\circ$ clockwise rotation about the origin transforms $(x, y)$ into $(-x, -y)$. Applying this to $X' = (-4, 4)$:

$X'' = (-(-4), -(4)) = (4, -4)$

Final Answer

The final coordinates of the image of point $X$ after the transformations are:

$\boxed{(4, -4)}$

Would you like a more detailed breakdown or further clarification? Here are related questions to deepen your understanding:

1. How do reflections over the $x$- and $y$-axes differ?
2. How would a $90^\circ$ clockwise rotation affect the coordinates?
3. What is the formula for $270^\circ$ counterclockwise rotation, and how does it relate to a $90^\circ$ clockwise rotation?
4. What happens to the coordinates when an object is reflected over the line $y = x$?
5. Can we use matrices to simplify the transformations? If so, how?

Tip:

When solving transformation problems, apply each step one at a time, carefully tracking the coordinates after each operation.